I am making a purse (which I intend to felt) for the first time and the instructions state:

Cast on 80 sts. Join and knit around.

What does that mean? How does one "join" their knitting? I tried to find a video, but didn't see one, so I thought you might be able to help.

Also, and this is a totally different question, I have a lovely yarn that says it gets 3sts/in on #11 needles. If I used a smaller needle, like an 8, would it be possible to get 4sts/in? Or should I just knit a swatch and see what happens?

On your second question - just do a swatch. It may end up being close to 4 st/". The other thing swatching will show you is if you like the look of the fabric when it's knit at more stitches per inch than recommemded.
As an example: you could use fingering weight yarn and knit on size 13 needles, but it wouldn't make a very nice sock.

To join your stitches for knitting in the round, just begin to knit on the first stitch that you cast on and keep going. You probably will be able to get 4sts/inch on the 8s, maybe even 9s or 10s. You do have to try it out and see what you get - everyone's tension is different.